Frobenius algebras and skein modules of surfaces in 3-manifolds

TL;DR

This paper constructs skein modules of surfaces in 3-manifolds using Frobenius algebras, providing a presentation theorem that relates algebraic and topological structures.

Contribution

It introduces a new skein module framework for surfaces in 3-manifolds based on Frobenius algebras and proves a presentation theorem linking algebraic generators and topological relations.

Findings

Skein module of the 3-ball is isomorphic to the Frobenius algebra's ground ring

Presentation theorem describes generators as incompressible surfaces with algebraic coloring

Relations are given by tubing geometry and algebraic relations

Abstract

For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We prove a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing geometry in the manifold and relations of the algebra.